Year 1



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Theory



What Is an Option:

The idea of options is certainly not new. Ancient Romans, Grecians, and Phoenicians traded options against outgoing cargoes from their local seaports. When used in relation to financial instruments, options are generally defined as a "contract between two parties in which one party has the right but not the obligation to do something, usually to buy or sell some underlying asset". Having rights without obligations has financial value, so option holders must purchase these rights, making them assets. This asset derives their value from some other asset, so they are called derivative assets. Call options are contracts giving the option holder the right to buy something, while put options, conversely entitle the holder to sell something. Payment for call and put options, takes the form of a flat, up-front sum called a premium. Options can also be associated with bonds (i.e. convertible bonds and callable bonds), where payment occurs in installments over the entire life of the bond, but this paper is only concerned with traditional put and call options.

Origins of Option Pricing Techniques:

Modern option pricing techniques, with roots in stochastic calculus, are often considered among the most mathematically complex of all applied areas of finance. These modern techniques derive their impetus from a formal history dating back to 1877, when Charles Castelli wrote a book entitled The Theory of Options in Stocks and Shares. Castelli's book introduced the public to the hedging and speculation aspects of options, but lacked any monumental theoretical base. Twenty three years later, Louis Bachelier offered the earliest known analytical valuation for options in his mathematics dissertation "Th‚orie de la Sp‚culation" at the Sorbonne. He was on the right track, but he used a process to generate share price that allowed both negative security prices and option prices that exceeded the price of the underlying asset. Bachelier's work interested a professor at MIT named Paul Samuelson, who in 1955, wrote an unpublished paper entitled "Brownian Motion in the Stock Market". During that same year, Richard Kruizenga, one of Samuelson's students, cited Bachelier's work in his dissertation entitled "Put and Call Options: A Theoretical and Market Analysis". In 1962, another dissertation, this time by A. James Boness, focused on options. In his work, entitled "A Theory and Measurement of Stock Option Value", Boness developed a pricing model that made a significant theoretical jump from that of his predecessors. More significantly, his work served as a precursor to that of Fischer Black and Myron Scholes, who in 1973 introduced their landmark option pricing model.

The Black and Scholes Model:

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of their work is a startlingly accurate option pricing model. Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.

[pic]

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6) Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

The Black and Scholes Model:

Delta:

[pic]

Delta is a measure of the sensitivity the calculated option value has to small changes in the share price.

Gamma:

[pic]

Gamma is a measure of the calculated delta's sensitivity to small changes in share price.

Theta:

[pic]

Theta measures the calcualted option value's sensitivity to small changes in time till maturity.

Vega:

[pic]

Vega measures the calculated option value's sensitivity to small changes in volatility.

Rho:

[pic]

Graphs of the Black and Scholes Model:

This following graphs show the relationship between a call's premium and the underlying stock's price.

The first graph identifies the Intrinsic Value, Speculative Value, Maximum Value, and the Actual premium for a call.

[pic]

The following 5 graphs show the impact of deminishing time remaining on a call with:

S = $48

E = $50

r = 6%

sigma = 40%

Graph # 1, t = 3 months

Graph # 2, t = 2 months

Graph # 3, t = 1 month

Graph # 4, t = .5 months

Graph # 5, t = .25 months

Graph #1

[pic]

Graph #2

[pic]

Graph #3

[pic]

Graph #4

[pic]

Graph #5

[pic]

Graphs # 6 - 9, show the effects of a changing Sigma on the relationship between Call premium and Security Price

S = $48

E = $50

r = 6%

sigma = 40%

Graph # 6, sigma = 80%

Graph # 7, sigma = 40%

Graph # 8, sigma = 20%

Graph # 9, sigma = 10%

Graph #6

[pic]

Graph #7

[pic]

Graph #8

[pic]

Graph #9

[pic]

Subject: Derivatives - Black-Scholes Option Pricing Model

Last-Revised: 5 Jan 2001

Contributed-By: Kevin Rubash (arr at bradley.edu)



The Black and Scholes Option Pricing Model is an approach for calculating the value of a stock option. This article presents some detail about the pricing model.

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of their work is a startlingly accurate option pricing model. Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.

The model is expressed as the following formula.

C = S * N(d1) - K * (e ^ -rt) * N (d2)

ln (S / K) + (r + (sigma) ^ 2 / 2) * t

d1 = --------------------------------------

sigma * sqrt(t)

d2 = d1 - sigma * sqrt(t)

Where:

C = theoretical call premium

S = current stock price

N = cumulative standard normal distribution

t = time until option expiration

r = risk-free interest rate

K = option strike price

e = the constant 2.7183..

sigma = standard deviation of stock returns (usually written as lower-case 's')

ln() = natural logarithm of the argument

sqrt() = square root of the argument

^ means exponentiation (i.e., 2 ^ 3 = 8)

(boy, HTML just isn't much good for formulas!)

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, K(e^-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

The Black and Scholes Model makes the following assumptions.

1. The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2. European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3. Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itt process. To understand what a continuous Itt process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itt process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4. No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that individual investors pay is more substantial and can often distort the output of the model.

5. Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6. Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

Volatility

Volatility is one of the most important factors in an option's price. It measures the amount by which an underlying asset is expected to fluctuate in a given period of time. It significantly impacts the price of an option's premium and heavily contributes to an option's time value. In basic terms, volatility can be viewed as the speed of change in the market, although you may prefer to think of it as market confusion. The more confused a market is, the better chance an option has of ending up in-the-money. A stable market moves slowly. Volatility measures the speed of change in the price of the underlying instrument or the option. The higher the volatility, the more chance an option has of becoming profitable by expiration. That's why volatility is a primary determinant in the valuation of options' premiums. There are options strategies that can be used to take advantage of either scenario.

For example, on the first Friday of each month, the government releases the employment report. As soon as the report is released, a fluctuation in the bond market usually occurs. This produces a simultaneous volatility crush in many stocks and their volatility increases. If a market's volatility was sitting just below 10, perhaps it is at 15 after the release of the report. You can equate that 5-point rise to an approximate 17% increase in options. Afterwards, volatility usually reverts to its normal levels. In addition, options often increase in price even if the price of the underlying stock doesn't move anywhere. Driven by volatility swings, event-driven situations offer profit-making opportunities if you know how to take advantage of them. There is a general rule of thumb: Buy options in low volatility; sell options during periods of high volatility. Markets with lots of volatility trigger an inflation of option prices. A market that moves a lot increases the probability that an option on that stock will end up in-the-money.

There are two basic kinds of volatility: implied and historical. Historical volatility is calculated by using the standard deviation of underlying asset price changes from close to close of trading going back 21 to 23 days. Implied volatility is a computed value that measures an option's volatility, rather than the underlying asset. The fair value of an option is calculated by entering the historical volatility of the underlying asset into an option pricing model (Black-Scholes for stocks). The computed fair value may differ from the actual market price of the option. Implied volatility is the volatility needed to achieve the option's actual market price. In more basic terms, historical volatility (also called statistical volatility) gauges price movement in terms of past performance and implied volatility approximates how much the marketplace thinks prices will move. Volatility is a very important piece of the puzzle, not only for analyzing an option's value, but for assessing a market's inclination for dramatic price movement.

In its most basic form, volatility means change. Since implied volatility is a computed value, it is best calculated with the aid of a computer that can easily match theoretical option prices with current market prices of the option. Volatility can be estimated without the use of a computer. Calculating an option's implied volatility is a very complicated process that uses options-pricing models to assume the future price movement of the underlying instrument. However, a trader can assess a stock's historical volatility by looking at its price range-the bigger the price range of a stock, the higher its volatility. The formula for calculating historical volatility is also quite complex; however, determining the price range for a stock over the relatively recent past (anywhere from 10 days to 12 months) can approximate historical volatility. To determine a stock's historical volatility, calculate the equilibrium level (midpoint) of a stock's price range. Then simply divide the difference between the high point and the equilibrium level by the equilibrium level to get the volatility percentage. For example, Ameritech Corp. has been fluctuating between 30 (support) and 50 (resistance) for the last year. The equilibrium level is 40 [(50-30) / 2 = 10 and 50 -10 = 40]. Ameritech's approximate volatility is 25% [(50-40) / 40 = 25%]. This percentage can then be used to forecast the maximum price action a stock is likely to make throughout the next 12 months.

Examining the difference between a stock's historical volatility and implied volatility can also help traders to recognize when a stock option is underpriced or overpriced. If the option's implied volatility is higher than the historical volatility, the option is theoretically overpriced. Option sellers look for these kinds of opportunities to sell high and buy low. In contrast, option buyers look for underpriced options by searching for market situations in which the implied volatility of an option is lower than the historical volatility (buy low and sell high). You may choose to spend a lot of time screening markets to determine their volatility. In general, there are two very different kinds of markets. The first one has plenty of movement, and high volatility. The second market has very low volatility. In general, sell stock and options with high volatility and buy stock and options with low volatility.

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